Question

Determine each of the following areas under the standard normal (z) curve: a. To the left of -1.28 b. To the right of 1.28 c. Between -1 and 2 d. To the right of 0 e. To the right of -5 f. Between -1.6 and 2.5 g. To the left of 0.23

Short answer

a. 0.1003, b. 0.1003, c. 0.8186, d. 0.5000, e. approx. 1, f. 0.9452, g. 0.5910.

Step-by-step solution

Understanding the z-score

A z-score measures how many standard deviations an element, X, is from the mean of the distribution, in this case, 0. Negative z-scores are below the mean, positive ones are above it. To interpret z-scores (which will be the values given in the exercise), one can use a Z-table or online calculator.

Calculate area under the curve

(a) To the left of -1.28, we look up this z-score in the z-table or use an online calculator. (b) To the right of 1.28, we calculate 1 minus the value for 1.28 from the z-table or online calculator. (c) Between -1 and 2, we subtract the z-score for -1 from the z-score for 2. (d) To the right of 0 is simply 0.5, as zero is exactly in the middle. (e) To the right of -5 is almost 1 as -5 is far away to the left under the curve. (f) Between -1.6 and 2.5, we subtract the z-score for -1.6 from that for 2.5. (g) To the left of 0.23, we simply use the value from the table or online calculator.

Use of Z-table or Online Calculator

For each part (a-g), we use a Z-table or online calculator to find the appropriate values. Students need to search the z-score and find the corresponding area under the curve.